1. **State the problem:** Kevin needs to choose between two loan offers for financing an expansion program. First Bank offers a loan with monthly payments at a quoted annual interest rate of 7.26%. Second Bank offers a loan with semi-annual payments at a quoted annual interest rate of 7.3%. We need to determine which loan is better by comparing their effective annual interest rates (EAR). 2. **Formula and explanation:** The effective annual rate (EAR) accounts for compounding periods and is given by: $$EAR = \left(1 + \frac{r}{n}\right)^n - 1$$ where $r$ is the nominal annual interest rate (as a decimal), and $n$ is the number of compounding periods per year. 3. **Calculate EAR for First Bank:** - Quoted rate $r = 0.0726$ - Compounding periods $n = 12$ (monthly) $$EAR_1 = \left(1 + \frac{0.0726}{12}\right)^{12} - 1$$ Calculate inside the parentheses: $$1 + \frac{0.0726}{12} = 1 + 0.00605 = 1.00605$$ Raise to the 12th power: $$1.00605^{12} \approx 1.0752$$ Subtract 1: $$EAR_1 = 1.0752 - 1 = 0.0752 = 7.52\%$$ 4. **Calculate EAR for Second Bank:** - Quoted rate $r = 0.073$ - Compounding periods $n = 2$ (semi-annual) $$EAR_2 = \left(1 + \frac{0.073}{2}\right)^2 - 1$$ Calculate inside the parentheses: $$1 + \frac{0.073}{2} = 1 + 0.0365 = 1.0365$$ Raise to the 2nd power: $$1.0365^2 = 1.0743$$ Subtract 1: $$EAR_2 = 1.0743 - 1 = 0.0743 = 7.43\%$$ 5. **Compare EARs:** - First Bank EAR = 7.52% - Second Bank EAR = 7.43% Since the Second Bank has a lower effective annual rate, Kevin should take the loan from the Second Bank. **Final answer:** Kevin should choose the loan from the Second Bank because it has a lower effective annual interest rate of 7.43% compared to First Bank's 7.52%.